Conventionally, lubricating oil has been used in an internal combustion engine, a transmission or other mechanical devices to facilitate the smooth operation thereof. In particular, a lubricating oil (engine oil) for an internal combustion engine is required to have a high level of performances because the internal combustion engine has been improved in performance, enhanced in output and used under severe working conditions. Therefore, conventional engine oils have been blended with various additives such as antiwear agents, metallic detergents, ashless dispersants, and anti-oxidants to meet such requisite performances (for example, see Patent Literatures 1 to 3 below). Furthermore, recently the fuel saving performance of the lubricating oil has been required to be increasingly better and better, and thus applications of a high viscosity index base oil or various friction modifiers have been studied (for example, see Patent Literature 4 below).